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Lectures on Scattering Theory LECTURES ON SCATTERING THEORY DMITRI YAFAEV Lecture notes prepared by Andrew Hassell, based on lectures given by the author at the Australian National University in October and November, 2001. The first two lectures are devoted to describing the basic concepts of scattering theory in a very compressed way. A detailed presentation of the abstract part can be found in [33] and numerous applications in [30] and [36]. The last two lectures are based on the recent research of the author. 1. Introduction to Scattering theory. Trace class method 1. Let us, first, indicate the place of scattering theory amongst other mathe- matical theories. This is very simple: it is a subset of perturbation theory. The ideology of perturbation theory is as follows. Let H0 and H = H0 +V be self-adjoint operators on a Hilbert space H, and let V be, in some sense, small compared to H0. Then it is expected that the spectral properties of H are close to those of H0. Typically, H0 is simpler than H, and in many cases we know its spectral family E0(·) explicitly. The task of perturbation theory is to deduce information about the spectral properties of H = H0 + V from those of H0. We shall always consider the case of self-adjoint operators on a Hilbert space H. The spectrum of a self-adjoint operator has two components: discrete (i.e., eigen- values) and continuous. Hence, perturbation theory has two parts: perturbation theory for the discrete spectrum, and for the continuous spectrum. Eigenvalues of H0 can generically be shifted under arbitrary small perturbations, but the formulas for these shifts are basically the same as in the finite dimensional case, dim H < ∞, which is linear algebra. On the contrary, the continuous spectrum is much more stable. Example. Let ∞ H0 = −∆,H = −∆ + v(x), v(x) = v(x), v ∈ L (R), (1.1) be self-adjoint operators on the Hilbert space H = L∈(R). Denote by Φ the Fourier transform. Then ∗ 2 H0 = Φ |ξ| Φ, (1.2) 2 so that the spectrum of H0 is the same as that of the multiplication operator |ξ| . Moreover, we know all functions of the operator H0 and, in particular, its spectral family explicitly. In the case considered V is multiplication by the function v. Thus, each of the operators H0 and V is very simple. However, it is not quite so simple to understand their sum, the Schr¨odinger operator H = H0 + V . Nevertheless, it is easy to deduce from the Weyl theorem that, if v(x) → 0 as |x| → ∞, then 1 2 DMITRI YAFAEV the essential spectrum of H is the same as that of H0 and hence it coincides with [0, ∞). Scattering theory requires classification of the spectrum in terms of the theory of measure. Each measure may be decomposed into three parts: an absolutely continuous part, a singular continuous part, and a pure point part. The same classification is valid for the spectral measure E(·) of a self-adjoint operator H. √√ Thus, there is a decomposition of the Hilbert space H = Hac ⊕ H∫c ⊕ H into the orthogonal sum of invariant subspaces of the operator H; the operator restricted to √√ Hac, H∫c or H shall be denoted Hac, Hsc or Hpp, respectively. The pure point part corresponds to eigenvalues. The singular continuous part is typically absent. Actually, a part of scattering theory is devoted to proving this for various operators of interest, but in these lectures we study only the absolutely continuous part Hac of H. The same objects for the operator H0 will be labelled by the index ‘0’. We ac denote P0 the orthogonal projection onto the absolutely continuous subspace H0 of H0. The starting point of scattering theory is that the absolutely continuous part of self-adjoint operator is stable under fairly general perturbations. However assump- tions on perturbations are much more restrictive than those required for stability of the essential spectrum. So scattering theory can be defined as perturbation theory for the absolutely continuous spectrum. Of course, it is too much to expect that ac ac H = H0 . However, we can hope for a unitary equivalence: ac ac ∗ ac ac H = UH0 U ,U : H0 → H onto. The first task of scattering theory is to show this unitary equivalence. We now ask: how does one find such a unitary equivalence U? This is related (although the relationship is not at all obvious) to the second task of scattering theory, namely, the large time behaviour of solutions u(t) = e−iHtf. of the time-dependent equation ∂u i = Hu, u(0) = f ∈ H. ∂t If f is an eigenvector, Hf = λf, then u(t) = e−iλtf, so the time behaviour is evi- dent. By contrast, if f ∈ Hac, one cannot, in general, calculate u(t) explicitly, but scattering theory allows us to find its asymptotics as t → ±∞. In the perturbation theory setting, it is natural to understand the asymptotics of u in terms of solutions of the unperturbed equation, iut = H0u. It turns out that, under rather general ac ± ac assumptions, for all f ∈ H , there are f0 ∈ H0 such that ± ± −iH0t ± u(t) ∼ u0 (t), t → ±∞, where u0 (t) = e f0 , or, to put it differently, −iHt −iH t ± lim e f − e 0 f = 0. (1.3) t→±∞ 0 ± Hence f0 and f are related by the equality f = lim eiHte−iH0tf ±, t→±∞ 0 which justifies the following fundamental definition given by C. Møller [23] and made precise by K. Friedrichs [8]. LECTURES ON SCATTERING THEORY 3 Definition 1.1. The limit iHt −iH0t W± = W±(H, H0) = s- lim e e P0, t→±∞ if it exists, is called the wave operator. ± It follows that f = W±f0 . The wave operator has the properties ac (i) W± is isometric on H0 . (ii) W±H0 = HW± (the intertwining property). ac In particular, H0 is unitarily equivalent, via W±, to the restriction H|ran W± of ac H on the range ran W± of the wave operator W± and hence ran W± ⊂ H . ac Definition 1.2. If ran W± = H , then W± is said to be complete. It is a simple result that W±(H, H0) is complete if and only if the ‘inverse’ wave operator W±(H0,H) exists. Thus, if W± exists and is complete (at least for one ac ac of the signs), H0 and H are unitarily equivalent. It should be emphasized that scattering theory is interested only in the canonical unitary equivalence provided by the wave operators. Another important object of scattering theory ∗ S = W+W− (1.4) is called the scattering operator. It commutes with H0, SH0 = H0S, which follows ac directly from property (ii) of the wave operators. Moreover, it is unitary on H0 if both W± are complete. In the spectral representation of the operator H0, the operator S acts as multiplication by the operator-valued function S(λ) known as the scattering matrix (see the next lecture, for more details). An important generalization of Definition 1.1 is due to Kato [19]. Definition 1.3. Let J be a bounded operator. Then the modified wave operator W±(H, H0,J) is defined by iHt −iH0t W±(H, H0,J) = s- lim e Je P0, (1.5) t→±∞ when this limit exists. Modified wave operators still enjoy the intertwining property W±(H, H0,J)H0 = HW±(H, H0,J), ac but of course their isometricity on H0 can be lost. 2. We have seen that the wave operators give non-trivial spectral information about H. Thus, it is an important problem to find conditions guaranteeing the existence of wave operators. There are two quite different methods: the trace class method, and the smooth method (see the next lecture). The trace class method is the principal method of abstract scattering theory. For applications to differential operators, both methods are important. The fundamental theorem for the trace class method is the Kato-Rosenblum theorem [15, 31, 16]. Recall that a compact operator T on H is in the class Sp, p > 0, if p X ∗ p/2 ||T ||p = λj(T T ) < ∞. In particular, S1 is called the trace class and S2 is called the Hilbert-Schmidt class. 4 DMITRI YAFAEV Theorem 1.4. If the difference V = H − H0 belongs to the trace class, then the wave operators W±(H, H0) exist. This is a beautiful theorem. It has a number of advantages, including: (i) Since the conditions are symmetric with respect to the operators H0 and H, the wave operators W±(H0,H) also exist and hence W±(H, H0) are complete. (ii) The formulation is simple, but all proofs of it are rather complicated. (iii) It relates very different sorts of mathematical objects: operator ideals, and scattering theory. (iv) It is effective, since it is usually easy to determine whether V is trace class. (v) It is sharp, in the sense that if H0 and p > 1 are given, there is a V ∈ Sp such that the spectrum of H0 + V is purely point. However, Theorem 1.4 has a disadvantage: it is useless in applications to dif- ferential operators. Indeed, for example, for the pair (1.1) V is a multiplication operator which cannot be even compact (unless identically zero), and therefore Theorem 1.4 does not work. Nevertheless, it is still useful if one is only interested ac ac in the unitary equivalence of H and H0 = H0 . In fact, it is sufficient to show −1 −1 that the operators (H +c) and (H0 +c) are unitarily equivalent for some c > 0 or, according to Theorem 1.4, that their difference is trace class.
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